From Toeplitz to Greene-Lobb

The Prompt

Explain the inscribed square problem to a reader who knows what a closed curve is. State the conjecture; note that it has been proved for convex and smooth curves but remains open for arbitrary continuous curves. Cite Toeplitz (1911), Schnirelmann (1944), and Greene and Lobb's 2020 result on rectangles of every aspect ratio. Explain the surprise that the proof of the smooth case required symplectic geometry — a tool from mathematical physics — and what this suggests about the depth hidden in a visually simple question.

Completion Criteria

A short essay (three or four paragraphs) with at least three citations — Toeplitz (1911), Schnirelmann (1944), and Greene-Lobb (2020/2021) — and a note on the role of symplectic geometry.

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